To use a rational number to describe how far a point on the number line is from 0, you can begin by partitioning the unit interval into an arbitrary number of equal parts. Each of those parts can then be partitioned into an arbitrary number of equal parts, and those, in turn, can be partitioned again.

This process is actually a composition of operations. You can use arrow notation to keep a record of your partitioning actions, as well as the size of the subintervals being produced.

For example, what if you wanted to locate 17/48 on a number line from 0 to 1? You would start by drawing the number line on a piece of paper and repeatedly folding it, making sure to mark the locations of 0 and 1 before you start folding:

Here's one set of partitioning actions to find 17/48:

Video SegmentIn this video segment, the participants place a fractional value on a number line using the method of partitioning. They explore the reciprocal relationship that exists between partitioning and the number of units in a measure.

Is there more than one way to do the partitioning to arrive at a particular fraction?

If you are using a VCR, you can find this segment on the session video approximately 2 minutes and 40 seconds after the Annenberg Media logo.

The compensatory principle states that the smaller the subunit you use to measure the distance, the more of those subunits you will need; the larger the subunit, the fewer you will need. When multiples of two different subunits cover the same distance, different fraction names result. There is only one rational number associated with a specific distance from 0, so these fractions are equivalent. Note 7

For example, when measuring the diameter of a pencil using two different subunits, we would have the following:

But 1/4 and 2/8 are equivalent fractions, so these are the same measurements.

Problem A8

State the compensatory principle in your own words. What type of relationship exists between the size of a measuring unit and the number of that unit needed to measure a property?